J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey. On Devaney's definition of chaos, Am. Math. Monthly, 99 (1992) 332-334.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....apart. We show in this subsection that the Z action generated by parabolic or hyperbolic elements of PSL(2, R) is chaotic. Theorem 7.6 Let # be a hyperbolic or parabolic element of PSL(2, R) regarded as a transformation on Har(D) Then # defines a chaotic dynamical system. 13 It is proven in [3] that the first two conditions in the definition of chaos imply the third, so we only need to verify that these Z actions are topologically transitive and have a dense set of periodic points. Let K n = z # 1 n and define for # Har(D) ### n=1 sup z#Kn #(z) This norm ....

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey. On Devaney's definition of chaos, Am. Math. Monthly, 99 (1992) 332-334.

....with corresponding valuation defined by 1 ) t . It is conventional to regard this exceptional place as the infinite one, and to write P# (k) t The next examples show how the valuations work in practice. The first is a polynomial and the second is a rational function. Example 2 [1] Let p = 7 and consider the polynomial f(t) t t 4. This may be factorized using standard methods (from Chapter 4 of [18] for example) into f(t) t 3) t t 3) t 5t 2) Each of the three factors is irreducible over F 7 (see Table C in the Appendix of [18] ....

....group of S integers R S in k, defined by The continuous group endomorphism # : X X is dual to the monomorphism # : R S R S defined by #(x) #x. To explain this definition and to show how it relates to cellular automata, consider the following examples. Example 3 [1] Let k = F p (t) S = #, and # = t. Then R S = F p [t] and so X = 1 = # p . The map # is therefore the full one sided shift on p symbols. Equivalently, the map # is the cellular automaton with one sided state space and with local rule f(x 0 , x 1 ) x 1 . and # = t. Recall ....

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J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey. On Devaney's definition of chaos. Amer. Math. Monthly 99, 332--334 (1992).

....be interpreted as an element of regularity Gamma Devaney [7] Delta , the sensitivity property captures the idea that in a chaotic system a very small change in the initial condition can cause a big change in the trajectory. Recently, the definition of Devaney was simplified by Banks et al. [1]. Indeed, these authors showed that any continuous and topologically transitive map T : X X whose periodic points are dense in X has sensitive dependence on initial conditions. Sensitive dependence on initial conditions is widely understood as being the central idea of chaos and was popularized ....

....that for any ffl 0, there exists j 0 such that for any A 2 B(X) A c ) j ) A c ) ffl. Since (A) diam A, we consequently have diam A (A) X) Gamma (A c ) X) Gamma ffl. Applying Theorem 3.2 with s(ffl) max(0; X) Gamma ffl) gives the desired conclusion. Xi Banks et al. [1] showed that any continuous and topologically transitive map T : X X whose periodic points are dense in X has sensitive dependence on initial conditions. In the 9 particular case where X = 0; 1] one can deduce from Theorem 3.2 the following corollary, which completes the result of Banks et ....

Banks, J., Brooks, J., Cairns, G., Davis, G. and Stacey, P. (1992) On Devaney's definition of chaos, American Mathematical Monthly, Vol. 99, pp. 332--334.

....points of f are dense in X; and (3) f has sensitive dependence on initial conditions. Condition (1) means that for all non empty open subsets U and V of X there exists a natural number n such that f n (U) V 6= In the sequel we shall dispense with condition (3) Indeed, it has been proven in [Ba al] that if f is transitive and has dense periodic points then f has sensitive dependence on initial conditions. One of the most appealing properties of Definition 1.1 is that it extends rather naturally to linear systems( McC] P A] D S W] More precisely, let X be an infinite dimensional ....

J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly 99 (1992), 332--334.

....ffi 0 such that for any x 2 X and for any neighborhood N(x) of x, there is a point y 2 N(x) and a natural number n, such that d(F n (x) F n (y) ffi: ffi is called sensitivity constant. Condition (3) is known as sensitive dependence to initial conditions or simply sensitivity. In [2] it has been proved that for general dynamical systems (1) and (2) imply (3) In [4] it has been proved that, for CA, transitivity alone implies sensitivity. Thus, for CA, the notion of transitivity becomes central to chaos theory. In order to apply the Devaney s definition of chaos to CA we use ....

....disconnected space and F is a (uniformly) continuous map. 3 Additive CA are chaotic In this section we prove the main result of the paper. More precisely, we show that additive one dimensional CA defined on a finite alphabet of prime cardinality satisfy conditions (1) and (2) above and thus, by [2], are chaotic in the sense of Devaney. We now prove that if a one dimensional CA is leftmost or rightmost permutive then it is transitive. Theorem 1 Let f be any local rule. If f is rightmost [leftmost] permutive then F is transitive. Proof. Assume, without loss of generality, that f is ....

J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On the Devaney 's Definition of Chaos. Amer. Math. Monthly, 332-334, 1992. 17

....as Definition 1 and 2. In Definition 2, the case that S is without interior points is a trivial case. We do not treat this special case. In [4] Devaney adds the third condition to Definition 2: c) The periodic points of F are dense in S. Remark 3. The conditions (b) and (c) imply condition (a) [1]. But the importance and relationship to chaos of condition (c) should not be neglected (see [25] Now we define a discrete infinite dimensional linear chaotic system. Let (X; d) be a Fr echet space over the complex field C (i.e. a complete linear metric space over C) We denote Sigma(X ) the ....

J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly , 99(1992),332-334.

....time and space, and a regular grid topology. The above mentioned definitions of chaos are interrelated. We can state the following claims. i) Mathematical chaos implies physical chaos, i.e. a dynamical system which is transitive and has dense periodic orbits, is sensitive to initial conditions [1]. In particular, for CA, this claim holds even for weaker definitions of mathematical chaos, e.g. transitivity alone implies sensitivity [4] ii) Mathematical chaos implies low complexity. A dynamical system which is chaotic according to the mathematical notion of chaos is not a universal model ....

J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On the Devaney 's Definition of Chaos. Amer. Math. Monthly, 332-334, 1992.

....conditions if there is a # 0suchthatforanyx#Xand for any neighborhood H x of x there is y # H x such that f n (x) f n (y) #. Devaney [1989] defined chaos as a system satisfying both properties above together with the requirement that the set of periodic points of f be dense in X. Banks et al. 1992] have shown that the topological transitivity combined with the density of periodic points already implies sensitivity to initial conditions. Consequently, Ingraham [1992] requires topological transitivity and sensitivity to initial conditions as the defining properties of chaos. Topological ....

Banks, J., Brooks, J., Cairns G., Davis, G. & Stacy P. [1992] "On Devaney's definition of chaos," Amer. Math. Monthly 99, April '92, 332--334.

....the natural extension of a point map (as is customary in topology) is to invite confusion. It is much better to begin with a closure operator, as one s paradigm because transformations can be wildly misbehaved. In fact, probably the best definition of chaos is given in terms of transformations [1]. There is ample scope for misbehavior because we observe that if U and U 0 are sets of n elements each, there exist only n n distinct functions f : U U 0 compared to (2 n ) 2 n transformations U f Gamma U 0 . To achieve any results of interest we must constrain the ....

J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey. On Devaney's definition of chaos. Amer. Math. Monthly, 99(4):332--334, Apr. 1992.

....sensitive systems defy numerical computation. A DTDS is transitive if it cannot be divided into two independent subsystems that do not interact each other. The above properties are not independent and are also linked with surjectivity: Proposition 1 Regularity and transitivity imply sensitivity [1]. If a CA is transitive or regular then it is surjective. This proposition justifies that surjectivity is necessary for chaoticity. Proof. The proof that transitivity imply surjectivity is in [7] Here we prove that if a CA is not surjective then it is also not regular. Let P be the set of ....

J. Banks, J. Brooks, G. Davis G. Cairns, and P. Stacey. On Devaney's definition of chaos. Am. Math. Monthly, 99:332--334, 1992.

....behavior (type S ) They evolve to configurations where a generalized alternating subshift behavior occurs. ffl CA with aperiodic behavior (type A ) This is the same class as type c of [6] but we prefer to call it aperiodic because it is not really chaotic in the sense of Devaney s definition [9, 10]. This phenomenon is also called spatio temporal chaos or intermittency . Phenomenologically, what we observe is a number of patterns growing, vanishing and moving towards the future. There is a kind of regularity (these forms are far from noise) but also a diversity (different forms) There is ....

Banks J., Brooks J., Cairns G., Davis G. & Stacey P., (1992). On devaney's definition of chaos. The American Mathematics Monthly, 99(4):332--334.

....the following three conditions, for a mapping F : E # E. c1) F has sensitive dependence on initial conditions (c2) F is topologically transitive (c3) periodic points are dense in E . It has been shown that the three conditions are not independent and can be reduced to two by eliminating (c1) [Ban92], or all three can be replaced with a single combined condition [Tou97] c0) each pair of nonempty open sets shares a periodic orbit. Nevertheless, the original three conditions are of interest because they have reasonable intuitive meanings. First, sensitive dependence. suggests future ....

J. Banks et al , "On Devaney's definition of chaos," Amer. Math. Monthly , v99#4, 1992.

....which are not numbers. The TDS (X; S) is topologically transitive if every open set is eventually moved all over X ; this is the topological analogue of ergodicity. X; S) is chaotic if it is topologically transitive and the periodic points of S are dense in X. This definition is equivalent [1] to the one given in the influential textbook [3] the two stated conditions imply a third condition, sensitivity to initial conditions, which some other authors take as the definition of chaos. The TDS (X; S) is topologically mixing if, given two subsets A; B of X, S eventually mixes them up . ....

J. Banks, et al., On Devaney's definition of chaos, Amer. Math. Monthly, April 1992, 332--334.

....on which it operates as follows: 8x 2 Sigma 1 ; f n (x) oe m (x) where n 2 N and m 2 Z, and oe : C C is the shift: 8i 2 Z; oe(x) i = x i 1 . It is possible to prove that this kind of behavior leads to chaos in the sense of Devaney (topological transitivity, density of periodic points) [5, 6]. When observing a specific cellular automaton starting from a random initial configuration, what we see is the initial configuration progressively shifting to the right or to the left, together with a kind of periodic behavior. If we take an initial finite configuration in a zero background, for ....

Banks J., Brooks J., Cairns G., Davis G. & Stacey P., (1992). On devaney's definition of chaos. The American Mathematics Monthly, 99(4):332--334.

....point in X, there exists a periodic point of the function, i.e. a point that cycles under iteration. Thus, the third condition for chaotic behavior guarantees the presence of regularity in the midst of the wild actions of the iterates described in the first two conditions. Banks, et al. in [B], prove that the first condition is implied by the other two. In [AG2] the authors give two counterexamples that show that the second and third conditions are not implied by the respective other conditions. Closed Set. A closed set, roughly speaking, is one that contains its boundary. See also ....

J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, On Devaney's definition of chaos, Amer. Math. Monthly 99 (1992), 332--4.

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J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, "On Devaney's Definition of Chaos," American Mathematical Monthly, Vol. 99, No. 4, 332-334. April 1992.

....a set of measure 0 do the iterates T x; T 2 x; T 3 x; converge to a cycle of finite length. Hence, for almost all x 2 Sigma, T n x neither goes to a fixed point or a limit cycle, so for most of Sigma the asymptotic behavior of T n x is complicated. The binary left shift map on [0,1] is ergodic with respect the Lebesgue measure [22] We can use the topological conjugacy relation (61) we derived in section 5 to prove that the renormalized matching pursuit map F ( 2) is also ergodic with respect to the measure (S) h(S) where h is the conjugacy relation from (61) when ....

J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, "On Devaney's Definition of Chaos," American Mathematical Monthly, Vol. 99, No. 4, 332-334. April 1992.

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J. Banks, J. Brooks, G. Cairns, G. Davis and P. Stacey, "On Devaney's definition of Chaos," The American Mathematical Monthly, Vol. 99, No.4, pp. 332-334, 1992.

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J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, "On Devaney's Definition of Chaos," American Mathematical Monthly, Vol. 99, No. 4, 332-334. April 1992.

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